Theorem ontric3g 43521

Description: For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)

Assertion

Ref	Expression
ontric3g	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ontric3g

Step	Hyp	Ref	Expression
1		orcom 870	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥))
2	1	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
3		onsseleq 6398	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
4		ontri1 6391	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
5	2, 3, 4	3bitr2d 307	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ 𝑦))
6	5	con2bid 354	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
7	6	ancoms 458	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
8	4	ancoms 458	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
9		ontri1 6391	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
10	8, 9	anbi12d 632	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥)))
11		eqss 3979	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦))
12		ioran 985	. . . 4 ⊢ (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥))
13	10, 11, 12	3bitr4g 314	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
14		equcom 2018	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
15	14	orbi2i 912	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦))
16	15	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
17		onsseleq 6398	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
18	16, 17, 9	3bitr2d 307	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ ¬ 𝑦 ∈ 𝑥))
19	18	con2bid 354	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))
20	7, 13, 19	3jca 1128	. 2 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥))))
21	20	rgen2 3185	1 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361

This theorem is referenced by: (None)